He is addicted to watching movies. Why the writer used gerund after the preposition "to"? , because as known, we use the infinitive after the prepostion (to).
To here is an ordinary preposition, not the infinitive marker. With words derived from the stem addict- it introduces the substance (literal or metaphoric) to which one is addicted:
The thief was addicted to heroin.
She has an addiction to 17th-century metaphysical poetry.
He is addicted to making snarky comments on ELL.
The -ing form is employed here as a 'gerund'—that is, a verbform which may act externally in most roles of an ordinary noun, including standing as the object of a preposition.